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Entanglement-assisted classical communication and transmission of a quantumsystem are the two quantum resources for information processing. Manyinformation tasks can be performed using either quantum resource. However, thisequivalence is not always present since entanglement assisted classicalcommunication is known to sometimes be the better performing resource. Here, weshow not only the opposite phenomenon; that there exists tasks for whichtransmission of a quantum system is a more powerful resource than entanglementassisted classical communication, but also that such phenomena can have asurprisingly strong dependence on the dimension of Hilbert space. We introducea family of communication complexity problems parametrized by dimension ofHilbert space and study the performance of each quantum resource. We find thatfor low dimensions, the two resources perform equally well, whereas fordimension seven and above, the equivalence is suddenly broken and transmissionof a quantum system becomes more powerful than entanglement assisted classicalcommunication. Moreover, we find that transmission of a quantum system may evenoutperform classical communication assisted by the stronger-than-quantumcorrelations obtained from the principle of Macroscopic Locality.

The Magical Number Seven: An Unexpected Dimensional Threshold in Quantum Communication Complexity